## The modified, discrete, Lévy-transformation is Bernoulli.(English)Zbl 0761.60043

Séminaire de probabilités XXVI, Lect. Notes Math. 1526, 157-161 (1992).
Summary: [For the entire collection see Zbl 0754.00008.]
From the absolute value of a martingale, $$X$$, there is a unique increasing process that can be subtracted so as to obtain a martingale, $$Y$$. Paul Lévy discovered that if $$X$$ is Brownian motion, $$B$$, then $$Y$$, too, is a Brownian motion. Equivalently, Lévy found that the transformation that maps $$B$$ to $$Y$$ is measure-preserving. Whether it is ergodic, a question raised by Marc Yor, is open. Here, the natural analogue of Lévy’s transformation for the symmetric random walk is modified and, thus modified, is shown to be measure-preserving. The ergodicity of this transformation is then established by showing that it is isomorphic to the one-sided, Bernoulli shift-transformation associated with a sequence of independent random variables, each uniformly distributed on the unit interval.

### MSC:

 60G48 Generalizations of martingales

Zbl 0754.00008
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